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Iterated integrals of Jacobi polynomials. (English) Zbl 1443.33024

The \(m\)th Jacobi polynomial with parameters \(\alpha,\beta\in\mathbb{R}\) is defined as \[P_{n}^{(\alpha, \beta)}(z)=\sum_{k=0}^{n}\binom{n+\alpha}{n-k}\binom{n+\beta}{k}\binom{2n+\alpha+\beta}{n}^{-1}(z-1)^{k}(z+1)^{n-k}.\] For a fixed \(m \in \mathbb{Z}_{+},\) let \(\mathcal{P}_{n, m}^{(\alpha, \beta)}\) be the monic polynomial of degree \(n+m\) given by \[\mathscr{P}_{n, m}^{(\alpha, \beta)}=P_{n+m}^{(\alpha-m, \beta-m)}.\] This polynomial is called the \(m\)th fundamental iterated integral of \(\frac{(n+m)!}{n!}P_n^{(\alpha, \beta)}\).
There is renewed interest in the location of zeros of these polynomials, because this knowledge can be applied to the numerical solution of differential equations. Motivated by these applications, the authors give several theorems about the location if zeros of the \(\mathscr{P}_{n, m}^{(\alpha, \beta)}\) polynomials. Among many other nice results, they prove that the union of the zero sets of all these polynomials can be limited by a compact set of the complex plane.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30E15 Asymptotic representations in the complex plane
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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References:

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